ODE
\[ x^3 y''(x)=a \left (x y'(x)-y(x)\right )^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0932135 (sec), leaf count = 25
\[\left \{\left \{y(x)\to -\frac {x \log \left (-\frac {a \left (c_2 x+c_1\right )}{x}\right )}{a}\right \}\right \}\]
Maple ✓
cpu = 0.041 (sec), leaf count = 23
\[ \left \{ -x{\it \_C1}+{\frac {x}{a}{{\rm e}^{-{\frac {ay \left ( x \right ) }{x}}}}}+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x^3*y''[x] == a*(-y[x] + x*y'[x])^2,y[x],x]
Mathematica raw output
{{y[x] -> -((x*Log[-((a*(C[1] + x*C[2]))/x)])/a)}}
Maple raw input
dsolve(x^3*diff(diff(y(x),x),x) = a*(x*diff(y(x),x)-y(x))^2, y(x),'implicit')
Maple raw output
-x*_C1+1/a*exp(-a/x*y(x))*x+_C2 = 0